In this respect, the steady states of the completely positive QMEs mentioned above, which are obtained without resorting to secular approximation, are of great interest. It was recently noted [1]} that any modification of the Redfield equation to achieve complete positivity inevitably changes the perturbative steady state from  the MFG state. In this paper, we explicitly evaluate the steady states of these Lindbladian QMEs perturbatively. We focus on two prominent examples: the universal Lindblad equation (ULE) [2]}, [3]} and truncated Lindblad equation (TLE) [4]}. We use the analytic continuation method of Ref. [5]} to evaluate the second-order population, as well as the coherence. We find that the Gibbs state of the system Hamiltonian is not a steady state, but the effect of coupling to the bath is included in the solution (unlike the case of the secular approximation). We explicitly compare the steady states of these QMEs with the MFG state. We find that although the steady states of these Lindbladian equations capture the effect of coupling to the bath, they differ from the MFG state. This can be regarded as a tradeoff for Lindbladian QMEs derived from the Redfield equation. To ensure that the coupling effect persists, as in the MFG state, complete positivity must be sacrificed and vice versa.
